The best algorithm seems to be to construct the generalization of the Delaunay Triangulation to $d$-dimensional Euclidean space; that yields a collection of empty hyper-balls that are defined via $d+1$ of the points.
From that collection of hyperballs the ones, whose center is outside the convex hull of their defining $k>=d+1$ points, are not inside the convex hull of the $n$ points and are not considered further.
Then one has to check, whether the radius of any of the remaining hyper-balls is at least $1$.
From the efficient construction of the Delaunay Triangulation in higher dimensions, it follows, that the answer to the question is yes.